aahhh just another one of Ramanujan's random identities that he divined from the minds of the gods. reading these has been a reliable way for mathematicians to humble themselves since 1900.
he often wouldn't (or wasn't able to) prove them. they were just assertions that, much more often than not, turned out to be accurate.
That’s very fascinating. I guess it still is the case for many people in the world, that paper is too expensive. Whereas for many of us it’s readily available in massive, one might even say infinite, quantities.
How would a mere mortal could possibly divine equations like this?
There is a method behind it. The formulas are derived from the study of relatively simple ODEs, in this example f'(x) = x.f(x) + 1 and g'(x) = x.g(x) − 1. The latter was solved 100 years prior by Jacobi. While the solutions are non-trivial, they are fairly compact, accessible to a high schooler. The presentation linked from the blog is pretty good at unveiling the magic.
This is to say that, with proper guidance, the kid down the street can too become as cool as Ramanujan.
"Ramanujan said that, throughout his life, he repeatedly dreamed of a Hindu goddess known as Namagiri. She presented him with complex mathematical formulas over and over, which he could then test and verify upon waking. Once such example was the infinite series for Pi:
Describing one of his many insightful math dreams, Ramanujan said:
"While asleep I had an unusual experience. There was a red screen formed by flowing blood as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of results in elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing...""
I think this kind of explanation irritates many people for some reason. As if things just can't be that way, there must be a repeatable, reproducible recipe that someone else could follow, and if they did, they would get the same results as Ramanujan.
I think maybe because it flies in the face of everyone that claims 10x developers are not possible. Or that some people are just genetically gifted and there's nothing you could possibly do that would get you within a hundred yards of them no matter how much hard work you put in, or how good your teachers were, etc. (thinking about John von Neumann here).
It's sort of depressing to think that there's almost a different species of man amongst us. It's undeniable though.
There have been quite a few such people in history. It's really humbling to know that Galois did what he did at just 20, Gödel was just a year older than me when he broke PM, Neumann could solve problems faster than what it took a person to write a program on the computer to solve the problem and the computer running.
While genetics play a huge role, nurture plays a large role as well [1].
Yes, it goes against the cult of Effort, "10000 hours of deliberate practice" and all that. But I believe you're 100% right: those at the absolute right end of the distribution are often gifted. They work hard, too, but that is akin to sharpening the knife. It is not the knife.
I'm reminded of a documentary on Tiger Woods on one of the Discovery channels where the voice over is going, "How did Tiger Woods become such a genius? Was it his family's Buddhist traditions?", and it went on to list a bunch of possible factors, all the while showing a clip of Tiger Woods playing golf with absolute perfection AT THE AGE OF FOUR! I had to laugh at the mental effort being expended to avoid the horrific conclusion that he was born with it.
It's also the case that genius alone isn't sufficient. One also needs hard work and, as Ramanujan illustrates, access to institutions that can publish+legitimize the work.
I also find it fun to consider that the population is 7x larger today than it was in 1800, and there's generally much better access to education across the board. Around 1900 you've got a bunch of absolute giants in mathematics: Hilbert and Ramanujan, for starters, but also Riemann and Lobachevsky (whose noneuclidean geometry work was basically Einstein minus the physics). So, suppose O(5) world-changing geniuses. On raw population alone, one would expect about 35 people of similar caliber to be active today. But also mathematics as a profession, access to phd programs, etc, is many times larger per capita than it would have been in 1900. So I would personally guess that there's likely to be a couple hundred people currently active in math of a similar caliber.
Bear in mind that if someone of Euler's level is operating today, you very likely wouldn't know unless you were in the field, and it might not necessarily be obvious even then.
If you cloned Euler, and magically gave the clone a new upbringing that resulted in the same math skills, Euler!clone wouldn't be becoming famous for proving e^iπ + 1 = 0, because he'd have learned that in high school like the rest of us. Instead, he might go off and do something like Terence Tao and hack away at the Twin Prime conjecture in a series of papers that require a PhD in mathematics just to understand the abstract. It's a lot harder to become famous that way, even if the work being done is much harder in some sense.
I don't want to diminish the genius of pa-h mathematicians, because Euler is still a legitimate genius by any measure, but part of the reason why he could get around the way he did is that he was metaphorically working in a field where he could pluck ripe fruit off the ground. Similar geniuses exist today, but even as geniuses they still need ladders to get to the fruit now, and that just takes more time.
I'm not lamenting this, celebrating it, or judging it... it just is the way it is.
Cool, I commented to the same effect, but your food-based metaphor brings to mind the notion that we have less lead in the water today, better health care, a handle on air pollution etc. p. p. I am doubting that as I write it, thinking of the hopelessness of CO2 reduction and all the things to come after we have already passed the point of no return.
And I wouldn't want to grow up be a giraf either,if climbing a tree works as well, or just shake it up.
Oh boy, these metaphors are useless.
Let's put it another way. Picking the low hanging fruit has just become either, and it is still necessary to get there. As you mention Terry Tao, to imply at the same time that we had never heard of him is rather disingenious. What is the point?
It has been noted that science, for lack of a better word, is becoming increasingly specialized on the individual level and that we have no polymaths today as it were. Maybe you are still correct insofar as we from in the midst of it cannot yet really tell what combination of skill will take the crown.
But then let's drop a few names, Peter Shor, Noam Chomsky, Frederik Kortlandt. Oh that's right, you never heard of Kortlandt, probably my favorite Indo-Europeanist.
Interesting thought. We are after all posting under a link to John C. Baez.
But the thought is ultimately naive. The pedastol we put these people on doesn't grow with the number of people alive. I don't know what model of a social network you would need to make that calcjlation. The measure of genious that should obviously be part of that theory, but I am sure such measure does not exist objectively.
Besides, I'd argue that with an evergrowing body of knowledge and tooling necessary to weild it, the requirements and constraints for a genious today are different from the times that you want to compare. For a measure and linear growth you would need a linear space to begin with.
Really though, I just wanted to say that there is a ton of smart people out there. Shoulders of giants and all. Which should be a humbling thought.
What cult of effort are you talking about? I think usually people outside a field underestimate both the talent required and the effort. Children can be even more obsessed than adults also and be completely laser focused.
> I think this kind of explanation irritates many people for some reason. As if things just can't be that way, there must be a repeatable, reproducible recipe that someone else could follow, and if they did, they would get the same results as Ramanujan.
Oh, there is an easy recipe : just emulate the universe as it was back then and observe Ramanujan. As we know, having a recipe doesn’t mean we have the resources to implement it.
There are several movies on Ramanujan as well as biographic books. Apart from the tragic path of is life, these material also depict him not only as gifted but as a compulsively working his maths. So that’s no wonder he would even dream about it. And of course, you can model that as the result of unconscious thoughts throwing the result of problems that were fed to the mind during awaken time.
So, the magic recipe is practice, practice, and practice even more.
> I think maybe because it flies in the face of everyone that claims 10x developers are not possible.
On what metrics? The ability to be able to throw impressive "magical" code and being able to work efficiently with the rest of a team will not necessarily come together, for example.
> Or that some people are just genetically gifted and there's nothing you could possibly do that would get you within a hundred yards of them no matter how much hard work you put in, or how good your teachers were, etc.
“Be yourself; everyone else is already taken.” is generally attributed to Oscar Wilde.
Every life is genetically gifted, but not all lives encounter the environment that enables to thrives this gift – unfortunately.
Also no one is perfect. Many aspect of human life are not only dependent on how performant you are individually, so being really "too ahead" can be actually a severe handicap – at least if it doesn’t come with equal excellence into convincing other to trust you. And so depression is not something you can expect to be out of the realm of gifted evoked here.
I think it’s just the case that some people are born with intuition or better yet everyone is but some people are born with intuition that is game changing. Ramanujan had a natural intuition for infinite series as Einstein did for relativity and even though their work would have eventually been discovered one way or another I believe their intuition does give the impression they did what would have taken many people to do in a short time in comparison.
yes, I think this is the likely explanation - your brain processes information when you are asleep and many times you may get insights from the unconscious this way in the morning.
In addition to hard work Ramanujan was exceptionally gifted, and likely stumbled on some mental processes at an early age that gave him an edge, and spent the rest of his life perfecting them.
Real shame that he passed so early, due to a disease that would have easily been avoidable or curable today. Imagine what else he would have been able to achieve if given another 10-30 years.
I for one find the fact that such people existed very exciting. It means that some possible configuration of our physiology allows for amazing capabilities. Maybe nurturing has to do with it too, but the point is that it is possible.
Maybe one day (in a distant future most likely) the general population will have access to such mental prowess. This would then be another kind of singularity!
...or the 4-year-old chess prodigy, or the 5-year-old composer.
As an amateur chess player, I don't find it depressing -- although more likely I use it as an excuse. I know that there are kids out there who started out with a significantly-higher ranking within a year of playing than I will ever get to as an adult. (That said, many those same kids will eventually go on to put in tens of thousands of more hours than me.)
I guess the part that's slightly depressing is that, even if I slog and slog, and raise my ranking up by a bit, it will always feel like winning by perspiration, and not by divine inspiration. But I think it's fine for me to say "this isn't my game, I will play it so long as it's fun, but I won't ever be a genius at it."
I've had times when during a dream, I've heard a melody, and when I woke up, I picked up an instrument and was able to replicate the melody I heard when dreaming. If I can come up with good sounding melodies in my sleep despite being a fairly unpracticed musician, I can only imagine what geniuses in their respective fields are able to figure out unconsciously! Obviously the presentation within the above described dream is fairly dramatic, but that might just have been his mind's way of making sure he remembered it when he woke up.
Reminds me of TempleOS, "created by American programmer Terry A. Davis, who developed it alone over the course of a decade after a series of prophetic episodes that he later described as a revelation from God."
he often wouldn't (or wasn't able to) prove them. they were just assertions that, much more often than not, turned out to be accurate.